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Green's Theorem


Green's theorem is a vector identity which is equivalent to the curl theorem in the plane. Over a region D in the plane with boundary partialD, Green's theorem states

 ∮_(partialD)P(x,y)dx+Q(x,y)dy=intint_(D)((partialQ)/(partialx)-(partialP)/(partialy))dxdy,
(1)

where the left side is a line integral and the right side is a surface integral. This can also be written compactly in vector form as

 ∮_(partialD)F·ds=intint_(D)(del xF)·da.
(2)

If the region D is on the left when traveling around partialD, then area of D can be computed using the elegant formula

 A=1/2∮_(partialD)xdy-ydx,
(3)

giving a surprising connection between the area of a region and the line integral around its boundary. For a plane curve specified parametrically as (x(t),y(t)) for t in [t_0,t_1], equation (3) becomes

 A=1/2int_(t_0)^(t_1)(xy^'-yx^')dt,
(4)

which gives the signed area enclosed by the curve.

The symmetric form above corresponds to Green's theorem with P(x,y)=-y/2 and Q(x,y)=x/2, leading to

A=intint_(D)dxdy
(5)
=intint_(D)((partialQ)/(partialx)-(partialP)/(partialy))dxdy
(6)
=∮_(partialD)(-y/2)dx+(x/2)dy
(7)
=1/2int_(t_0)^(t_1)(-yx^'dt+xy^'dt)
(8)
=1/2int_(t_0)^(t_1)(xy^'-yx^')dt.
(9)

However, we are also free to choose other values of P and Q, including P(x,y)=0 and Q(x,y)=x, giving the "simpler" form

 A=int_(t_0)^(t_1)xy^'dt,
(10)

and P(x,y)=y and Q(x,y)=0, giving

 A=-int_(t_0)^(t_1)yx^'dt.
(11)

A similar procedure can be applied to compute the moment about the x-axis using P=-y^2/2 and Q=0 as

 M_x=intintydxdy=-1/2∮y^2dx=-1/2int_(t_0)^(t_1)y^2x^'dt
(12)

and about the y-axis using P=0 and Q=x^2/2 as

 M_y=intintxdxdy=1/2∮x^2dx=1/2int_(t_0)^(t_1)x^2y^'dt,
(13)

where the geometric centroid x^_=(x^_,y^_) is given by x^_=M_y/A and y^_=M_x/A.

Finally, the area moments of inertia can be computed using P=-y^3/3 and Q=0 as

 I_(xx)=intinty^2dxdy=-1/3∮y^3dx=-1/3int_(t_0)^(t_1)y^3x^'dt,
(14)

using P=-xy^2/2 and Q=0 as

 I_(xy)=intintxydxdy=-1/2∮xy^2dx=-1/2int_(t_0)^(t_1)y^2xx^'dt,
(15)

and using P=0 and Q=x^3/3 as

 I_(yy)=intintx^2dxdy=1/3∮x^3dy=1/3int_(t_0)^(t_1)x^3y^'dt.
(16)

See also

Area, Area Moment of Inertia, Curl Theorem, Divergence Theorem, Geometric Centroid, Multivariable Calculus, Stokes' Theorem

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References

Arfken, G. "Gauss's Theorem." §1.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 57-61, 1985.Kaplan, W. "Green's Theorem." §5.5 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 286-291, 1991.

Cite this as:

Weisstein, Eric W. "Green's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GreensTheorem.html

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